The Weyl group enjoys a number of interesting properties :
This close relationship between Coxeter groups and Kac–Moody algebras is the reason for denoting both with the same notation (for instance, denotes at the same time the Coxeter group with Coxeter graph of type and the Kac–Moody algebra with Dynkin diagram ).
Note that different Kac–Moody algebras may have the same Weyl group. This is in fact already true for finite-dimensional Lie algebras, where dual algebras (obtained by reversing the arrows in the Dynkin diagram) have the same Weyl group. This property can be seen from the fact that the Coxeter exponents are related to the duality-invariant product . But, on top of this, one sees that whenever the product exceeds four, which occurs only in the infinite-dimensional case, the Coxeter exponent is equal to infinity, independently of the exact value of . Information is thus clearly lost. For example, the Cartan matrices
Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the Kac–Moody context, the fundamental region is called “the fundamental Weyl chamber”.
We also note that by (standard vector space) duality, one can define the action of the Weyl group in the Cartan subalgebra , such that
Finally, we leave it to the reader to verify that when the products are all , then the geometric action of the Coxeter group considered in Section 3.2.4 and the geometric action of the Weyl group considered here coincide. The (real) roots and the fundamental weights differ only in the normalization and, once this is taken into account, the metrics coincide. This is not the case when some products exceed 4. It should be also pointed out that the imaginary roots of the Kac–Moody algebras do not have immediate analogs on the Coxeter side.
As the first (respectively, second) Cartan matrix defines the Lie algebra (respectively ) introduced below in Section 4.9, we also write it as (respectively, ). We denote the associated sets of simple roots by and , respectively. In both cases, the Coxeter exponents are , , and the metric of the geometric Coxeter construction is
We associate the simple roots with the geometric realisation of the Coxeter group defined by the matrix . These roots may a priori differ by normalizations from the simple roots of the Kac–Moody algebras described by the Cartan matrices and .
Choosing the longest Kac–Moody roots to have squared length equal to two yields the scalar products
Recall now from Section 3 that the fundamental reflections have the following geometric realisation
We now want to compare this geometric realisation of with the action of the Weyl groups of and on the corresponding simple roots and . According to Equation (4.56), the Weyl group acts as follows on the roots
while the Weyl group acts as
We see that the reflections coincide, , , , as well as the scalar products, provided that we set , , and . The Coxeter group generated by the reflections thus preserves the lattices
It follows, of course, that the Weyl groups of the Kac–Moody algebras and are the same,
and its symmetrization
The Weyl group of the corresponding Kac–Moody algebra is isomorphic to the Coxeter group above since, according to the rules, the Coxeter exponents are identical. But the action is now
and cannot be made to coincide with the previous action by rescalings of the ’s. One can easily convince oneself of the inequivalence by computing the eigenvalues of the matrices , and with respect to .
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